The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proof. This is a special case of Limits, Proposition 31.11.2. $\square$

Proof for a finite order thickening. Suppose that $X \subset X'$ is a finite order thickening with $X$ affine. Then we may use Serre's criterion to prove $X'$ is affine. More precisely, we will use Cohomology of Schemes, Lemma 29.3.1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_{X'}$-module. It suffices to show that $H^1(X', \mathcal{F}) = 0$. Denote $i : X \to X'$ the given closed immersion and denote $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp : \mathcal{O}_{X'} \to i_*\mathcal{O}_ X)$. By our discussion of finite order thickenings (following Definition 36.2.1) there exists an $n \geq 0$ and a filtration

\[ 0 = \mathcal{F}_{n + 1} \subset \mathcal{F}_ n \subset \mathcal{F}_{n - 1} \subset \ldots \subset \mathcal{F}_0 = \mathcal{F} \]

by quasi-coherent submodules such that $\mathcal{F}_ a/\mathcal{F}_{a + 1}$ is annihilated by $\mathcal{I}$. Namely, we can take $\mathcal{F}_ a = \mathcal{I}^ a\mathcal{F}$. Then $\mathcal{F}_ a/\mathcal{F}_{a + 1} = i_*\mathcal{G}_ a$ for some quasi-coherent $\mathcal{O}_ X$-module $\mathcal{G}_ a$, see Morphisms, Lemma 28.4.1. We obtain

\[ H^1(X', \mathcal{F}_ a/\mathcal{F}_{a + 1}) = H^1(X', i_*\mathcal{G}_ a) = H^1(X, \mathcal{G}_ a) = 0 \]

The second equality comes from Cohomology of Schemes, Lemma 29.2.4 and the last equality from Cohomology of Schemes, Lemma 29.2.2. Thus $\mathcal{F}$ has a finite filtration whose successive quotients have vanishing first cohomology and it follows by a simple induction argument that $H^1(X', \mathcal{F}) = 0$. $\square$


Comments (3)

Comment #1089 by Nuno Cardoso on

When is a finite order thickening this result is an easy consequence of Serre's criterion for affineness. Since 31.11.2 does not seem to be an easy result, I wonder if it would be worthy to add the direct proof of this particular case here.

Comment #1094 by on

Thanks for this and your other comments. I have added a proof for the finite order case, see this commit.

There are many places where one can give easier arguments in special cases, or just easier arguments. Submissions of this kind are very welcome!

In the general set-up of the Stacks project it would make sense to have a separated lemma for the finite order case and move it to the section containing Serre's criterion with a forward link to this general result... This is a TODO.

Comment #3819 by slogan_bot on

Suggested slogan: Affineness is insensitive to thickenings

There are also:

  • 2 comment(s) on Section 36.2: Thickenings

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